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PQRS is a square. So PQ = RS
Since X and Y are the mid-points of RS and PQ respectively,
The diagonal of the square SQ and XY bisect each other at O.
So XO = YO

\angle SOX= \angle QOY\ (opposite angles)
\angle OXS= \angle OYQ\ (both are right angles)

So ΔSOX \cong ΔYOQ
Congruent triangles have equal area. 
So ar(ΔSOX) = ar(ΔYOQ) (proved)
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